## 7. Discussion## 7.1. Implications for the underlying 3-dimensional structureThe result from Sect. 5, namely that an ensemble of randomly
positioned clumps with a given power law mass spectrum and a given
mass-size relation has a Nevertheless we can derive certain constraints for the 3-dim
structure from the fact that the observed 2-dimensional projection has
Along this line of reasoning, the 3-dimensional structure thus also
has a power law For a full understanding of the 3-dim cloud structure a link to the
velocity structure is crucial. The velocity structure is important not
only as a tool to tell overlapping 3-dim spatial structure apart, and
to keep the emission of individual clumps from mutual shielding in the
radiative transfer. Physically, the velocity structure must be linked
to the density structure via the magneto-hydrodynamic equations
describing the turbulent internal cloud motions. Ultimately, a proper
description of this magneto hydrodynamic turbulence should describe
the cloud structures observed. We are obviously far from reaching this
goal. A simple approach, assuming independent ## 7.2. Observational limitsThe fact that detailed observations of molecular cloud structure,
covering a significant dynamic range in spatial scales and also in the
signal to noise ratio within a reasonable integration time, have
become possible only within the last decade is linked to rapid
progress in technology. The high sensitivity of large mm-wave
telescopes and the excellent low noise performance of SIS-heterodyne
receivers available at the low- It is thus of particular importance to estimate the observing time needed for extension of a cloud map to higher angular resolution with an interferometer array. Let us thus consider for simplicity and consistency the situation, where we want to observe a single resolution element of a large scale cloud map with higher angular resolution using an interferometer of antennas identical to the single dish used for the low angular resolution map. The effective system temperature (including receiver sensitivity, telescope efficiency and atmospheric losses) for each interferometer element and the single dish telescope are assumed to be identical. The noise level reachable in a given integration time and resolution bandwidth with the interferometer scales as (see Downes 1989) where is the noise achieved with the single
dish telescope in the same given integration time, The interferometer observes all resolution
elements within its field of view simultaneously. The single dish
telescope has to observe each resolution element separately for a time
. The scaled up single dish map with the same
number of pixels as the interferometer map thus takes a total time
. For the interferometer map to be a true
scaled down version of the larger map, the signal to noise ratio per
resolution element has to be the same as for the single dish map. The
scaling of the signal level with resolution is given by the assumed
Combining the above relations and solving for equal signal to noise at the resolution of both maps gives the scaling of the total integration times: The same equation can be derived by considering the increase in
integration time with higher resolution according to
due to the decrease of
the signal level for an otherwise identical, but larger single dish
telescope; the reduced efficiency of an We see that the total integration time rapidly increases with increasing resolution, . This increase is compensated by the decrease of integration time in proportion to the number of interferometer baselines. With the value of derived above, and assuming a 5-element interferometer such as the IRAM Plateau-de-Bure instrument, i.e. , a submap of a single resolution element of the large scale single dish map with the interferometer at a 10 times higher resolution would thus take about 60 times as long as the total large scale map. This is unrealistically long to persue, considering the already very long observing times needed for decent size single dish maps with adequate signal to noise ratio. Extending studies of cloud structures to higher angular resolution will thus be feasible only with very large () future interferometer arrays. ## 7.3. Beyond fractional Brownian motion structureThe present paper discusses molecular cloud structure in the
framework of what is commonly called "monofractal structure". The
actual cloud structure is certainly more complex and many different
structures with the same mono-fractal characteristics are possible. A
few recent papers have attempted to characterise observed clouds with
multifractal methods. They show that the cloud structure indeed shows
multifractal properties, both in the velocity distribution (Miesch
& Bally 1994) and in the column density distribution (Chappell
& Scalo 1997). However, they also seem to indicate that these
multifractal properties vary from cloud to cloud. Whether this is due
to the limited data base available and possible systematic errors in
the analysis, or whether observed molecular clouds in fact fall into a
single "universality class" is not clear at the moment. Molecular
clouds clearly need more than one single parameter, e.g. a single
fractal dimension, a single © European Southern Observatory (ESO) 1998 Online publication: July 20, 1998 |